Operations of Functions! (Division) And some other stuff...

Question

How would you solve this f(x)⁄g(x) problem (please include ALL steps and an explanation for each)?

 

f(x)=(5x+11)/(x²-9)

g(x)= (4x-2)/(x²-9)

 

 

Also, would the domain end up being (-∞, -3) U (-3, 1/2) U (1/2, 3) U (3, ∞)? Or would there be no 1/2? From what I think I got, the denominator should be 2(2x-1), so the denominator couldn't be 0, so it can't be 1/2. Am I right? Or would we just be looking in the middle of the solving/simplifying?

 

One last thing: when do we look in problems like these to get our domain (like when do we look to see what values are not included)? 

 

Thanks!

Answer 1

"... would the domain end up being (-∞, -3) U (-3, 1/2) U (1/2, 3) U (3, ∞)?"

Yes, it would!!

If you have variable in denominator, we must indicate the domain of a giving expression, or equation, or function!!!

So, the first step will be to find the domain of f(x) and g(x),

x² - 9 ≠ 0 ----> x ≠ ± 3.

Thus, both functions are defined for

x ∈ ( - ∞, - 3) U (- 3, 3) U (3, ∞)

........

After you finish division, you will insert another critical point x = 1/2 into domain of original functions.   

Answer 2

Hi M;

I APOLOGIZE.  I FORGOT ABOUT THE SECOND HALF OF THE EQUATION.  HERE IT IS...

f(x)/g(x)=f(x) times 1/g(x)

(5x+11)    (x²-9)

(x²-9)       (4x-2)

 

I took g(x) and flipped the numerator and denominator to make this a multiplication rather than division question.

 

In this equation (x²-9) is in both the numerator and denominator.  These therefore cancel.

 

(5x+11)

(4x-2)

 

THE ONLY THING LIMITING THE CURRENT DOMAIN IS THAT X CANNOT = 1/2 TO RENDER THE NUMERATOR ZERO.

 

PLEASE ASK YOUR INSTRUCTOR IF WE SIMPLIFY AND THEN EVALUATE DOMAIN, OR IF WE EVALUATE DOMAIN BEFORE SIMPLIFICATION.  I HAVE RESEARCHED THIS AND CANNOT FIND AN ANSWER.